# Feynman-Kac formulae

January 27, 2021 — January 27, 2021

Bayes

Monte Carlo

probabilistic algorithms

probability

state space models

statistics

swarm

time series

There is a mathematically rich theory about particle filters, and the central tool to make that go seems to be *Feynman-Kac formulae*. The notoriously abstruse Del Moral (2004);Doucet, Freitas, and Gordon (2001) are regarded as the unifying introduction to these formulae, whatever they are. A diligent student will supposedly make consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos” and their delicate relationships. I will get around to understanding them myself eventually, maybe?

Related, apparently: Backward SDEs.

## 1 References

Beck, E, and Jentzen. 2019. “Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.”

*Journal of Nonlinear Science*.
Cérou, Moral, Furon, et al. 2011. “Sequential Monte Carlo for Rare Event Estimation.”

*Statistics and Computing*.
Chan-Wai-Nam, Mikael, and Warin. 2019. “Machine Learning for Semi Linear PDEs.”

*Journal of Scientific Computing*.
Chopin, and Papaspiliopoulos. 2020.

*An Introduction to Sequential Monte Carlo*. Springer Series in Statistics.
Chuang. 2010. “The Feynman-Kac Formula:Relationships Between Stochastic DifferentialEquations and Partial Differential Equations.”

Del Moral. 2004.

*Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications*.
Del Moral, and Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.”

*The Annals of Applied Probability*.
Del Moral, Hu, and Wu. 2011.

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Del Moral, and Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In

*Séminaire de Probabilités XXXIV*. Lecture Notes in Mathematics 1729.
Doucet, Freitas, and Gordon. 2001.

*Sequential Monte Carlo Methods in Practice*.
Doucet, Godsill, and Andrieu. 2000. “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering.”

*Statistics and Computing*.
E, Han, and Jentzen. 2017. “Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.”

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Han, Jentzen, and E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.”

*Proceedings of the National Academy of Sciences*.
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*Entropy*.
Hutzenthaler, Jentzen, Kruse, et al. 2020. “Overcoming the Curse of Dimensionality in the Numerical Approximation of Semilinear Parabolic Partial Differential Equations.”

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*.
Hutzenthaler, and Kruse. 2020. “Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities.”

*SIAM Journal on Numerical Analysis*.
Kebiri, Neureither, and Hartmann. 2019. “Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations.” In

*Stochastic Dynamics Out of Equilibrium*. Springer Proceedings in Mathematics & Statistics.
Nualart, and Schoutens. 2001. “Backward Stochastic Differential Equations and Feynman-Kac Formula for Lévy Processes, with Applications in Finance.”

*Bernoulli*.
Nüsken, and Richter. 2021. “Solving High-Dimensional Hamilton–Jacobi–Bellman PDEs Using Neural Networks: Perspectives from the Theory of Controlled Diffusions and Measures on Path Space.”

*Partial Differential Equations and Applications*.
Papanicolaou. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.”

*arXiv:1504.05309 [Math, q-Fin]*.